Illumination of Convex Bodies with Symmetries in Dimensions 3 and 4

  • Author / Creator
    Sun, Wen R
  • Let n ≥ 3 and B ⊂ ℝⁿ. The Illumination Conjecture states that the minimal number I(B) of directions/‘light sources’ that illuminate the boundary of a convex body B, which is not the affine image of a cube, is strictly less than 2ⁿ. The conjecture in most cases is widely open, and it has only been verified for certain special classes of convex bodies. For instance, significant progress in dimension 3 has been made for convex bodies with certain symmetries. Moreover, in large dimensions with dimension greater than some universal constant C, Konstantin Tikhomirov showed the conjecture in the setting of 1-symmetric bodies, but unfortunately there is a gap in his proof which still leaves a case to be handled. In addition, an explicit value for this constant C was not computed. The natural question which follows from Tikhomirov’s paper is whether we can use Tikhomirov’s method for 1-symmetric bodies or 1-unconditional bodies in low dimensions. In this thesis, we will first fill the gap in Tikhomirov’s results. Through this process, we are also able to prove the Illumination Conjecture, with bound 2ⁿ (and not < 2ⁿ), for any 1-symmetric convex body regardless of dimension. Additionally, we are able to show that for 1-symmetric convex bodies in dimensions 3 and 4, I(B) ≤ 7 and I(B) ≤ 15, respectively. Finally, we are also able to show that for 1-unconditional polytopes in dimensions 3 and 4, I(B) ≤ 6 and I(B) ≤ 16, respectively.

  • Subjects / Keywords
  • Graduation date
    Fall 2022
  • Type of Item
  • Degree
    Master of Science
  • DOI
  • License
    This thesis is made available by the University of Alberta Library with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.